Civil Engineering Reference
In-Depth Information
If the concept of a linear acceleration is adopted (see Fig. 9.10.c), then
τ
()
(
)
r
τ
=+
r
r
−
r
(9.184)
k
k
+
1
k
t
Δ
in which case
τ
2
⎫
τ
τ
⎡
⎤
()
(
)
(
)
∫
d
r
τ
=+
r
r
+
r
−
r
τ
=+ +
r
r
τ
r
−
r
⎪
⎢
⎥
k
k
k
+
1
k
k
k
k
+
1
k
t
2
t
Δ
Δ
⎣
⎦
⎪
⎬
0
(9.185)
τ
⎡
2
⎤
2
3
τ
τ
τ
⎪
()
(
)
(
)
∫
d
r
τ
=+
r
rr
+ +
τ
r
−
r
τ
=+ +
rr r
τ
+
r
−
r
⎢
⎥
⎪
k
k
k
k
1
k
k
k
k
k
1
k
+
+
2
t
2
6
t
Δ
Δ
⎢
⎥
⎣
⎦
⎭
0
and thus
t
t
Δ
Δ
⎫
(
)
(
)
t
r
=+Δ +
r
r
r
− =+
r
r
r
+
r
⎪
⎪
⎬
k
1
k
k
k
1
k
k
k
k
1
+
+
+
2
2
(9.186)
2
2
2
2
t
t
t
t
Δ
Δ
Δ
Δ
⎪
(
)
r
=+Δ⋅
r
t
r
+
r
+
r
− =+Δ⋅
r
r
t
r
+
r
+
r
k
1
k
k
k
k
1
k
k
k
k
k
1
+
+
+
⎪
⎭
2
6
3
6
The concept of integrating an assumed variation of the acceleration between
t
and
t
presented above may all be generalised into the formulation first suggested by Newmark
[32]:
+
1
(
)
1
t
t
r
=
r
+
−
γ
⋅Δ⋅
r
+ ⋅Δ⋅
γ
r
⎫
k
1
k
k
k
1
+
+
⎪
⎬
1
2
(9.187)
⎛
⎞
2
2
r
=+Δ⋅
r
t
r
+ − ⋅ Δ⋅
β
t
r
+⋅ Δ⋅
β
t
r
⎜
⎟
⎪
k
1
k
k
k
k
1
+
+
⎝
⎠
⎭
where
are weighting parameters, each to be chosen according to prescribed
requirements regarding numerical stability and accuracy. From the second expression in
Eq. 9.187 the acceleration at
and
β
γ
t
1
+
1
⎡
1
1
⎤
⎛
⎞
(
)
r
=
r
−
r
−
r
+
−
1
⋅
r
(9.188)
⎢
⎜
⎟
⎥
k
1
k
1
k
k
k
+
+
2
t
2
β
Δ
β
β
Δ
t
⎝
⎠
⎣
⎦
is obtained, which, combined with the first expression in Eq. 9.187, renders
γ
⎛
γ
⎞ ⎛
γ
⎞
(
)
(9.189)
r
=
r
−
r
−
−
1
⋅
r
−
−
1
Δ ⋅
t
r
⎜
⎟ ⎜
⎟
k
1
k
1
k
k
k
+
+
t
2
β
Δ
β
β
⎝
⎠ ⎝
⎠
